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basic hypergeometric functions

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1: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
2: 17.18 Methods of Computation
The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …
3: 17.1 Special Notation
§17.1 Special Notation
The main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ϕ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) , Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) , Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) , and Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) . …
4: 17.4 Basic Hypergeometric Functions
§17.4 Basic Hypergeometric Functions
17.4.1 ϕ s r + 1 ( a 0 , a 1 , a 2 , , a r b 1 , b 2 , , b s ; q , z ) = ϕ s r + 1 ( a 0 , a 1 , , a r ; b 1 , b 2 , , b s ; q , z ) = n = 0 ( a 0 ; q ) n ( a 1 ; q ) n ( a r ; q ) n ( q ; q ) n ( b 1 ; q ) n ( b s ; q ) n ( ( 1 ) n q ( n 2 ) ) s r z n .
17.4.2 lim q 1 ϕ s r + 1 ( q a 0 , q a 1 , , q a r q b 1 , , q b s ; q , ( q 1 ) s r z ) = F s r + 1 ( a 0 , a 1 , , a r b 1 , , b s ; z ) .
17.4.3 ψ s r ( a 1 , a 2 , , a r b 1 , b 2 , , b s ; q , z ) = ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) = n = ( a 1 , a 2 , , a r ; q ) n ( 1 ) ( s r ) n q ( s r ) ( n 2 ) z n ( b 1 , b 2 , , b s ; q ) n = n = 0 ( a 1 , a 2 , , a r ; q ) n ( 1 ) ( s r ) n q ( s r ) ( n 2 ) z n ( b 1 , b 2 , , b s ; q ) n + n = 1 ( q / b 1 , q / b 2 , , q / b s ; q ) n ( q / a 1 , q / a 2 , , q / a r ; q ) n ( b 1 b 2 b s a 1 a 2 a r z ) n .
5: 17.15 Generalizations
§17.15 Generalizations
For higher-dimensional basic hypergometric functions, see Milne (1985a, b, c, d, 1988, 1994, 1997) and Gustafson (1987).
6: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
17.5.1 ϕ 0 0 ( ; ; q , z ) = n = 0 ( 1 ) n q ( n 2 ) z n ( q ; q ) n = ( z ; q ) ;
17.5.5 ϕ 1 1 ( a c ; q , c / a ) = ( c / a ; q ) ( c ; q ) .
7: Bibliography
  • G. E. Andrews (1966a) On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.
  • G. E. Andrews (1974) Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.
  • 8: 17.6 ϕ 1 2 Function
    17.6.17 ϕ 1 2 ( a , b c / q ; q , z ) ϕ 1 2 ( a , b c ; q , z ) = c z ( 1 a ) ( 1 b ) ( q c ) ( 1 c ) ϕ 1 2 ( a q , b q c q ; q , z ) ,
    17.6.18 ϕ 1 2 ( a q , b c ; q , z ) ϕ 1 2 ( a , b c ; q , z ) = a z 1 b 1 c ϕ 1 2 ( a q , b q c q ; q , z ) ,
    17.6.21 b ( 1 a ) ϕ 1 2 ( a q , b c ; q , z ) a ( 1 b ) ϕ 1 2 ( a , b q c ; q , z ) = ( b a ) ϕ 1 2 ( a , b c ; q , z ) ,
    9: 17.9 Further Transformations of ϕ r r + 1 Functions
    17.9.4 ϕ 1 2 ( q n , b c ; q , z ) = ( c / b ; q ) n ( c ; q ) n ( b z q ) n ϕ 2 3 ( q n , q / z , q 1 n / c b q 1 n / c , 0 ; q , q ) ,
    17.9.5 ϕ 1 2 ( q n , b c ; q , z ) = ( c / b ; q ) n ( c ; q ) n ϕ 2 3 ( q n , b , b z q n / c b q 1 n / c , 0 ; q , q ) .
    17.9.8 ϕ 2 3 ( q n , b , c d , e ; q , q ) = ( d e / ( b c ) ; q ) n ( e ; q ) n ( b c d ) n ϕ 2 3 ( q n , d / b , d / c d , d e / ( b c ) ; q , q ) ,
    17.9.9 ϕ 2 3 ( q n , b , c d , e ; q , q ) = ( e / c ; q ) n ( e ; q ) n c n ϕ 2 3 ( q n , c , d / b d , c q 1 n / e ; q , b q e ) ,
    10: 18.33 Polynomials Orthogonal on the Unit Circle
    18.33.15 ϕ n ( z ) = = 0 n ( a q 2 ; q 2 ) ( a ; q 2 ) n ( q 2 ; q 2 ) ( q 2 ; q 2 ) n ( q 1 z ) = ( a ; q 2 ) n ( q 2 ; q 2 ) n ϕ 1 2 ( a q 2 , q 2 n a 1 q 2 2 n ; q 2 , q z a ) ,
    For the notation, including the basic hypergeometric function ϕ 1 2 , see §§17.2 and 17.4(i). …